This account of Multilinear Algebra has developed out of lectures which I gave at the University of Sheffield during the session 1981/2. In its present form it is designed for advanced undergraduates and those about to commence postgraduate studies. At this general level the only special prerequisite for reading the whole book is a familiarity with the notion of a module (over a commutative ring) and with such concepts as submodule, factor module and homomorphism.

Multilinear Algebra arises out of Linear Algebra and like its antecedent is a subject which has applications in a great many different fields. Indeed, there are so many reasons why mathematicians may need some knowledge of its concepts and results that any selection of applications is likely to disappoint as many readers as it satisfies. Furthermore, such a selection tends to upset the balance of the subject as well as adding substantially to the required background knowledge. It is my impression that young mathematicians often acquire their knowledge of Multilinear Algebra in a rather haphazard and fragmentary fashion. Here I have attempted to weld the most commonly used fragments together and to fill out the result so as to obtain a theory with an easily recognizable structure.

The book begins with the study of multilinear mappings and the tensor, exterior and symmetric powers of a module. Next, the tensor powers are fitted together to produce the tensor algebra of a module, and a similar procedure yields the exterior and symmetric algebras.